Topology of a holomorphic vector field around an isolated singularity
Let \(F = \sum_{i=1}^ nF_ i (z) \partial/ \partial z_ i\) be a holomorphic vector field with isolated singularity at the origin \(0 \in \mathbb{C}^ n\) defined in an open set \(U\) containing \(\{0\}\), let \({\mathcal F}\) be the foliation over \(U \backslash \{0\}\) defined by the integral curves of \(F\), and \(\langle F,z \rangle = \sum_{i=1}^ n F_ i(z) \overline z_ i\) be the Hermitian scalar product. The authors consider a real-analytic vector field \(r_ F = - \overline {\langle F,z \rangle} \cdot F\) and call it the radial vector field. Suppose that all singularities of \(r_ F\) are nondegenerate, in other words, the vector field \(F\) has Morse type. Under these hypotheses the authors prove that the singularity set \(M = \{z\in \mathbb{C}^ n:\langle F,z \rangle = 0\}\) of the radial vector field is either a real-analytic variety of codimension two having the only singularity at \(0\in\mathbb{C}^ n\) or coincides with the origin. In the former case the set \(M\backslash\{0\}\) consists of a finite number of connected components and it is transversal to the fibration \({\mathcal F}\). Let \({\mathcal F}_ \varepsilon\) be the foliation defined by the intersection of fibres of \({\mathcal F}\) with the sphere \(S_ \varepsilon\) of the radius \(\varepsilon>0\) centered at the origin. The distance between the origin and points on fibres defines the function on \({\mathcal F}\). Assume also that this function has no saddle points on \({\mathcal F}\). Then it is proved that \({\mathcal F}_ \varepsilon\) is stable over \(S_ \varepsilon\), that is, for any \(S_{\varepsilon'}\) with \(\varepsilon'<\varepsilon\) there is a homeomorphism taking \(S_ \varepsilon\) into \(S_{\varepsilon'}\) and carrying the foliation \({\mathcal F}_ \varepsilon\) into \({\mathcal F}_{\varepsilon'}\).
scientific article; zbMATH DE number 652154